The Four Methods of Visualizing .9999... = 1

(And A Proof)

To Continue Directly to the Mathmematical Proof, Click Here

Method One: Fractional / Decimal Equivalance and Addition







Method Two: Algebraic Reasoning




Method Three: Logic and Conceptual Thinking

If it was not equal to one then there would be a number between it and 1. What number would that be?

It is intuitive to think: No matter how many 9's
you add, you'll never get all the way to 1. 

But that's how it seems if you think about moving toward 1.  What if 
you think about moving away from 1?

That is, if you start at 1, and try to move away from 1 and toward 
0.99999..., how far do you have to go to get to 0.99999... ?  Any step 
you try to take will be too far, so you can't really move at all -  
which means that to move from 1 to 0.99999..., you have to stay at 1.

When you think of 0.999... as being 'a little below 1', it's because 
in your mind, you've stopped expanding it; that is, instead of 

  0.999999...

you're really thinking of

  0.999...999

which is not the same thing. You're absolutely right that 0.999...999 
is a little below 1, but 0.999999... doesn't fall short of 1 until 
you stop expanding it. But you never stop expanding it, so it never 
falls short of 1.

Can you prove that .9999... does not equal 1?

Method 4: Arithmatic

Subtraction


Division

The Actual Proof

Infinate Geometric Series